Let `veca = 2hati+lambda_1 hatj+3hatk`, `vec(b)=4hati+(3-lambda_2) hatj+6hatk` and `vec(c)=3hati+6hatj+(lambda_3-1)hatk` be three vectors such that `vec(b)=2 vec(a)` and `vec(a)` is perpendicular to `vec(c)` then a possible value of `((lambda)_1,(lambda)_2,(lambda)_3)` is:

(a) `(1,3,1)`
(b) `((-1/2),4,0)`
(c) `(1,5,1)`
(d) `((1/2), 4, -2)`

Options

A. `(1/3, 4/3, 2/3)`
B. `(1/2 , 0 , - 1)`
C. `(4, - 3, 1)`
D. `(1/3 , 1/3 , 2/3)`

Answer

D

Explanation

`veca = hati + lamda_1 hatj = hatk , vec b = 2 hati + (1 - lamda_2 ) hatj + 2hatk , vec c = hati + hatj + (lamda_3 - 2) hatk `
` vecb = 2 veca , " " 1 - lamda_2 = 2 lamda_1 , " " 2 lamda_1 + lamda_2 = 1`
Also , `veca.vec c = 0 rArr 1 + lamda_1 + lamda_3 - 2 = 0`
`lamda_1 + lamda_3 = 1`
hence a possible value of ` (lamda_1, lamda_2,lamda_3) " is " (1/3, 1/3, 2/3)`